We obtain prompt diphoton cross section in proton-nucleus collisions in Hamiltonian light-cone approach within a hybrid approximation, treating the projectile proton in the parton model and the target nucleus in the color-glass-condensate approach. We study in detail the diphoton correlations in quark-nucleus and proton-nucleus collisions at RHIC and the LHC. We show that the single fragmentation diphoton produces the away-side correlations peak, and the double-fragmentation component of prompt diphoton is responsible for the near-side peak, and the long-range in rapidity near-side azimuthal collimation, the so-called "ridge" structure. We study the transverse momentum, density and energy dependence of the diphoton ridge and show that it strongly depends on the kinematics and saturation dynamics. We show that while diphoton ridge exists at the LHC in quark-nucleus collisions, the effect disappears in proton-nucleus collisions at the LHC. At RHIC the ridge-type structure persists at low transverse momenta of diphoton even in proton-nucleus collisions. We argue that diphoton correlation measurements in p + A collisions can help to discriminate among models and understand the true origin of the observed dihadron ridge in p + A collisions. We also show that in addition to the ridge structure, prompt diphoton correlation also exhibits some distinct novel features, including the emergence of away-side double-peak structure at intermediate transverse momenta.